Questions

Question 1

Recently, a new strain of Coronavirus (COVID X) was introduced to Canada. the strain behaves differently than earlier strains, and much like at the start of the COVID pandemic, no one has any immunity. Everyone is suspectible.

The number of infected individuals doubles every week, and it is projected to keep rising at this fixed rate. This growth rate is due to the fact that a novel virus has entered a population in which none of the individuals have ever come into contact with this specific virus strain before. A graphical representation of the number of COVID X cases over time is plotted below. What type of growth can be observed from the plot?

If you were a member of a health committee tasked with minimizing the number of infections, how could you reduce the rate of disease transmission? After implementing your suggestion, how would this alter the curve in the plot?

Question 2

A population of rabbits was introduced to the University of Waterloo campus in 2015. The president explained that “We need to mitigate the negative impacts of loneliness on campus. Our geese on campus need to interact with more species”. Seemed weird. Anyway, since rabbits have many overlapping generations per year, we can use the exponential growth equation: \[N_t=N_0 e^{rt},\]

answer the following questions:

  1. If a population of rabbits has a population of 75 in 2015 and 25 in 2020, solve for r.
  2. What would the population be in 2023?
  3. In what year will the population be 100 000?
  4. How will the level of goose loneliness change?

Question 3

An invasive species Common buckthorn (Rhamnus cathartica) has been filling up the woodlots and wild spaces of Ontario that used to be occupied by native trees https://www.invadingspecies.com/invaders/plants/common-buckthorn-2/. It produces large numbers of seeds once a year that germinate quickly, and then forms dense stands that prevent the growth of native trees and shrubs. If that wasn’t enough, this species can host oat rust, a fungus that causes leaf and crown rust and affects the yield and quality of oats, and the soybean aphid, an insect that damages soybean crops. Because it can affect agricultural crops, common buckthorn is listed as a noxious weed under Ontario’s Weed Control Act.

You notice a big buckthorn tree filled with berries in a 1 hectare woodlot near your house, nearby are 5 smaller trees that all look to be one year old. If you take no action, how many buckthorn trees do you expect to see 3 years from now?

Common buckthorn, photo by R.A.Nonenmacher, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

Question 4

You have been studying a population of Daphnia pulex, a type of zooplankton, for a few years.

You know that the Daphnia population grows exponentially and initially had a per capita intrinsic growth rate, r, of 1.2. You then noticed that although the population continues to grow exponentially, the growth rate has decreased to 0.36. If r continues to be 0.36, will the population size start decreasing? Explain why or why not.

Question 5

A new beetle species, Harrya Pottertae, has recently been discovered in the English countryside. Quite magically, researchers have noticed that the population has been increasing rapidly with overlapping generations. The population of the beetles was initially 13 individuals per metre squared and has grown to about 934 individuals per metre squared in the span of 8 days.

  1. Is this an example of exponential or geometric growth rate?

  2. What is the instantaneous rate of increase for this new beetle species? Remember \(e^x=y\) is equal to \(\ln y=x\).

Answers

Answer 1

The plot above is depicting exponential growth, which is sometimes referred to as a “J-shaped” curve or a “Hockey Puck” curve.

One possible solution to curbing the exponential growth of COVID X is vaccination. Vaccination has been shown to prevent epidemics by interrupting the spread of the disease even if some members of the population remain unvaccinated. This occurs through a phenomenon known as herd immunity, where the spread of the disease in a population becomes significantly limited after a minimum proportion of the population has been vaccinated.

After vaccination, the fixed rate of transmission, which is characteristic of exponential growth of diseases, would decrease. Since the rate of transmission is given by the slope of the curve at any given point, the curve would begin to plateau as the transmission rate decreases.

Answer 2

  1. Substituing population size in 2015 and 2020 into our equation for exponential growth we have: \[ \begin{aligned} 215&=75e^{r(2020-2015)} \\ \frac{215}{75}&=e^{r(5)} \\ \ln\frac{215}{75}&=r(5) \\ r&=0.21 \end{aligned} \] Therefore, r is 0.21\(\frac {individuals}{individual*year}\).

  2. The year 2023 is eight years after the introduction in 2015, so we have: \[ \begin{aligned} N_{2023}&=75e^{0.211\times8}\\ N_{2023}&=406 \end{aligned} \] Therefore, the population size would be 406 individuals in 2023.

  3. To find when the population will be 100,000 we need to solve for time as: \[ \begin{aligned} 100,000=75e^{0.211t}\\ \frac{100,000}{75}=e^{0.211t}\\ \ln\frac{215}{75}=0.211t\\ t=32.8 \end{aligned} \] Therefore, in approximately 33 years after 2015, around the year 2047, the population will be 100,000.

  4. Rabbits and geese both eat grass, and there is no evidence that goose enjoy the company of rabbits. We might naively expect the level of goose angriness to increase exponentially as the rabbit population grows, but it more likely the population growth of both species will slow as resources become limited, and the exponential model will no longer be appropriate.

Answer 3

Since berries are produced only once a year, this is an example of geometric population growth. We can predict future population size using the equation: \(N_t=\lambda N_0^t\), where is the annual population growth rate. We don’t have a lot of data, but let’s try!

Current population size = 6/hectare, and clearly started from the one tree. We don’t really know how old that tree is but, let’s just assume it is two years old, so that t = 1 year. Rearranging and substituting in we an solve for \(\lambda\) as: \[ \begin{aligned} N_2&=\lambda N_1^t \\ \lambda&= \frac{N_2}{N_1^t}\\ \lambda&=\frac{6}{1^1} \\ \lambda&=6 \end{aligned} \] Now that we have the population growth rate, we can use it to predict population size 3 years from now, if our starting population size is 6 individuals/hectare.

\[ \begin{aligned} N_{3}&=\lambda N_0^3 \\ N_{3}&=6(6^3) \\ N_{3}&=1296 \end{aligned} \] Gulp!! That’s 1296 trees in a one hectare woodlot!! Better take some action!

Answer 4

No, the population size will not decrease because r is still greater than 0. Recall that for exponential growth, the population size increases when r > 0, decreases when r < 0, and remains the same when r = 0. Therefore, because r is smaller than it was initially but is still greater than 0, the population size will now grow more slowly, but will not start decreasing.

*Daphnia Pulex*. Yale Peabody Museum, CC0, via Wikimedia Commons

Daphnia Pulex. Yale Peabody Museum, CC0, via Wikimedia Commons

Answer 5

  1. Since the population is increasing with overlapping generations, and the question is asking for an instantaneous rate of increase, this is an example of exponential growth rather than geometric growth.

  2. The equation for exponential growth rate is \(N_t=N_0 e^{rt}\) where \(N_0\) is 13 individuals per $m2, \(N_t\) is 934 individuals per \(m^2\), and t is 8 days.

  1. Rearrange the equation to solve for r. \[r=\frac{\ln\frac{N_t}{N_0}}{t}\]

  2. Sub in values and solve:

\[ \begin{aligned} r&=\frac{\ln\frac{934}{13}}{8}\\ r&=0.53 \end{aligned} \] Therefore, the instantaneous rate of increase is 0.53 \(\frac {individuals}{individual*year}\), and we expect a dramatic increase in the magic levels in the area. Read about their non-magical relatives here https://www.thecanadianencyclopedia.ca/en/article/beetle